there is a continues function f(x) on [a,b]

assuming that every sub section [alpha,beta]

which belongs to [a,b]

there is an integral in interval (on top: beta on the bottom:alpha) f(x)dx>0

prove that f(x)>=0 for every x on [a,b]

does it necessary f(x)>0 x on [a,b]

if its correct prove

if not show a disproving example

??